Near-Extremal and Extremal Stability

Exact extremality changes horizon dynamics: vanishing surface gravity permits conserved horizon quantities and polynomial growth of transverse derivatives. That instability can coexist with decay of the exterior metric, so “stable” and “unstable” are incomplete without the controlled quantity. As of 2026-08-20, nonlinear spherical extremal Reissner–Nordström has a substantial theorem cell and independent charged-scalar simulations expose a singular interior endpoint, while generic extremal Kerr remains open. The record is partially resolved and supported at moderate confidence.

Helpful background. Mode, Linear, and Nonlinear Stability Interfaces supplies the hierarchy of stability claims and the role of horizon modes. This record isolates the exact- and near-extremal endpoints.

Which extremal stability question is finite?

For a named extremal black hole, ask whether small perturbations keep the exterior close to a stationary family, whether the final state remains extremal or becomes subextremal, and which transverse horizon derivatives stay bounded or grow. The metric, scalar field, connection, curvature, and horizon energy can have different answers.

The axes are exact versus near extremality; Reissner–Nordström versus Kerr; spherical Einstein–Maxwell–neutral-scalar versus charged scalar or vacuum gravity; fixed-background versus backreacting evolution; codimension-one extremal tuning versus generic data; and the event horizon versus the region just inside it. Taking a near-extremal limit after long time need not equal evolving exactly extremal data.

Assessment at 2026-08-20	Meaning
Question status	Partially resolved
Evidence status	Supported
Confidence	Moderate across the record because the nonlinear RN theorem is symmetry and codimension restricted, while generic nonlinear Kerr remains open
Domain	Asymptotically flat extremal and near-extremal RN/Kerr cells, with field content and backreaction explicit
What could change the conclusion	A nonlinear generic evolution theorem or controlled endpoint calculation

A starting project: recover the conserved horizon quantity

Fit and preparation. This is a useful bridge from perturbation theory to mathematical relativity. You should be able to derive the scalar wave equation in ingoing horizon-regular coordinates and distinguish a transverse derivative from a derivative along the horizon. The analytic baseline needs only pencil and paper; a spherical fixed-background solver provides an optional second check.

Reading and baseline. Read Aretakis 2011, §12.1 (PDF) for the spherical conservation law. Derive it directly by restricting the regular wave equation to the extremal horizon. Specify the radial coordinate and normalization, and compare initial data with nonzero and vanishing conserved quantity. The zero-charge case is a useful control; absence of this leading diagnostic does not establish boundedness of every higher derivative.

Deliverable. Produce the derivation and, if computing, a resolution study of conservation along the horizon for smooth crossing data. Report horizon extraction error separately from wave-evolution error. This reproduces a linear statement on a fixed geometry. It neither measures gravitational backreaction nor demonstrates that an evolving black hole remains extremal.

Possible extension and stopping point. After checking later work on transient instability, choose a fixed scalar model and compare exact extremality with a sequence of small positive surface gravities. A candidate research question is how a declared finite-time diagnostic depends on both observation time and distance from extremality. The simple conservation-law baseline is already known; any extension requires a novelty check. Stop if the transverse derivatives fail to converge or numerical horizon drift is comparable with the effect. Coupling the scalar to gravity, tuning a nonlinear extremal endpoint, or moving to gravitational Kerr perturbations needs specialist supervision and a new set of constraints and invariant diagnostics.

What is established in the stated domain

Aretakis proves a conserved spherical horizon quantity Aretakis 2011, Part I, §12.1 (PDF), and nondecay of a transverse derivative with polynomial blow-up of higher derivatives Aretakis 2011, Part II, §2, Theorem 6 (PDF). These are fixed-background scalar-wave results on exact extremal Reissner–Nordström, not conclusions about metric destruction.

Angelopoulos, Kehle, and Unger prove nonlinear asymptotic stability of extremal Reissner–Nordström on a codimension-one stable manifold in spherical Einstein–Maxwell–neutral-scalar theory Angelopoulos, Kehle, and Unger 2026, Theorems 3.7 and 3.9, pp. 29–32; proof §§4–9. Their result combines exterior decay and convergence with a nonlinear Aretakis hierarchy. It shows directly that horizon derivative growth can coexist with nonlinear metric stability. The journal article was published online on 14 January 2026, before this record’s evidence cutoff.

Gelles and Pretorius independently simulate spherical Einstein–Maxwell–charged-scalar threshold evolutions that tune to a dynamical extremal horizon Figs. 2–11 and convergence analysis. They find divergent horizon energy and growing curvature immediately inside. The matter coupling and critical family differ from the analytic theorem, so the results provide complementary endpoint evidence rather than replication.

For extremal Kerr, Bishoyi, Sabharwal, and Khanna numerically support nonaxisymmetric gravitational horizon hair on a fixed background Bishoyi et al. 2025, numerical evolutions and charge/Ori-coefficient comparison. Backreaction and a generic nonlinear Kerr theorem remain absent. Dafermos’s 2025 perspective explicitly identifies those gaps §§3–5 and conjectures.

Results and open questions at 2026-08-20

Claim or evidence	What it supports	What it does not establish
Aretakis linear theorem	Exact extremal RN carries conserved horizon charge and derivative blow-up	Nonlinear backreaction
Angelopoulos–Kehle–Unger theorem	Nonlinear metric stability plus Aretakis hierarchy on a spherical stable manifold	Generic data or Kerr rotation
Gelles–Pretorius simulations	A dynamically tuned charged endpoint with horizon/interior singular growth	Vacuum Kerr or untuned generic evolution
Fixed-background extremal Kerr numerics	Nonaxisymmetric gravitational horizon hair	Coupled nonlinear stability
Current synthesis	Exact extremality is a tuned threshold with major Kerr gaps	A new theorem

The strongest conclusion is coexistence: exterior settling and transverse derivative instability are not opposites. The broad endpoint remains model dependent.

Competing interpretations of Aretakis growth

One interpretation calls any unbounded transverse derivative an instability. That is mathematically correct for the chosen norm and physically relevant when curvature or energy diverges. Its weakness is that it can suggest the exterior metric leaves every small neighborhood, which the nonlinear spherical theorem denies.

The stability interpretation focuses on decay of radiative fields and convergence of the metric to an extremal member. Its weakness is the opposite: it can hide a loss of regularity on or inside the horizon. A useful verdict must report both norms and state whether extremality was tuned teleologically or arose from a generic open set.

Counterexamples, null results, and failed routes

The nonlinear RN theorem is direct adverse evidence to the slogan “Aretakis growth destroys every extremal black hole.” The same theorem is adverse evidence to the slogan “settling makes all horizon derivatives harmless.” Neither conclusion transfers automatically to extremal Kerr.

Near-extremal holes have nonzero surface gravity and may show a long transient rather than an exactly conserved Aretakis quantity. Extrapolating finite-time near-extremal numerics to the exact endpoint is therefore unreliable unless tuning distance, resolution, and observation time are taken in a controlled double limit.

What would change the assessment

A nonlinear theorem for generic, nonsymmetric extremal Kerr perturbations—controlling exterior radiation, parameter drift, and invariant horizon curvature—would decide the largest open cell. A converged backreacting simulation with generic nonaxisymmetric data could materially constrain the endpoint if it separates exact growth from near-extremal transients. For RN, extending beyond the codimension-one spherical neutral-scalar manifold would enlarge the theorem’s relevance.

Research checklist

• Define the extremality diagnostic and whether it is imposed, tuned, or dynamically approached.
• Separate exact and near-extremal limits, and state the order of the long-time and tuning limits.
• Name field content, symmetry, backreaction, and the horizon-regular coordinate or tetrad used.
• Track metric decay, horizon charge, transverse derivatives, energy, and invariant curvature separately.
• For numerics, pin tuning algorithm, resolution hierarchy, constraints, horizon finder, and fit window.
• Do not infer nonlinear Kerr behavior from a fixed-background Teukolsky evolution.

Source and data map

Aretakis supplies the foundational fixed-background theorem. Angelopoulos–Kehle–Unger supplies the published nonlinear spherical theorem. Gelles–Pretorius supplies an independent backreacting charged-scalar endpoint; Bishoyi–Sabharwal–Khanna supplies Kerr-specific fixed-background evidence; Dafermos supplies the current problem taxonomy. Extremal AdS branes were excluded.

No end-to-end rerun was performed. No version-pinned public code and initial-data archive was verified for the numerical studies, and no machine-checkable proof archive was found. Reproduction requires a controlled extremality diagnostic, invariant horizon extraction, convergence and constraints, and a time window long enough to distinguish polynomial growth from a transient.

References

• Angelopoulos, Yannis, Christoph Kehle, and Ryan Unger. “Nonlinear Stability of Extremal Reissner–Nordström Black Holes in Spherical Symmetry.” Forum of Mathematics, Pi 14 (2026), e3, 1–84. DOI.
• Aretakis, Stefanos. “Stability and Instability of Extreme Reissner–Nordström Black Hole Spacetimes for Linear Scalar Perturbations I.” Communications in Mathematical Physics 307 (2011), 17–63. DOI. Open PDF.
• Aretakis, Stefanos. “Stability and Instability of Extreme Reissner–Nordström Black Hole Spacetimes for Linear Scalar Perturbations II.” Annales Henri Poincaré 12 (2011), 1491–1538. DOI. Open PDF.
• Bishoyi, Som Dev, Subir Sabharwal, and Gaurav Khanna. “Numerical Evidence for Non-Axisymmetric Gravitational ‘Hair’ for Extremal Kerr Black Hole Spacetimes with Hyperboloidal Foliations.” General Relativity and Gravitation 57 (2025), article 46. DOI.
• Dafermos, Mihalis. “The Stability Problem for Extremal Black Holes.” General Relativity and Gravitation 57 (2025), article 60. DOI.
• Gelles, Zachary, and Frans Pretorius. “Nonlinear Endpoint of Charged Horizon Instabilities.” Physical Review D 113 (2026), 124067. DOI.